The 3-rd unramified cohomology for norm one torus

TL;DR

This paper computes the third unramified cohomology of the norm one torus associated with an abelian extension, building on previous estimates for algebraic tori and providing explicit calculations.

Contribution

It provides an explicit computation of the third unramified cohomology for the norm one torus over an abelian extension, extending prior theoretical estimates.

Findings

Explicit formula for third unramified cohomology of norm one torus

Extension of previous estimates to specific class of tori

Enhanced understanding of cohomological invariants in algebraic tori

Abstract

For an algebraic torus $S$, Blinstein and Merkurjev have given an estimate of $3$-th unramified cohomology $\bar{H}^3_{nr}(F(S),\mathbb Q/\mathbb Z(2))$ obtained from a flasque resolution of $S$. Based on their work, for the norm one torus $W=R_{K/F}^{(1)}\mathbb{G}_m$ with $K/F$ abelian, we compute the $3$-th unramified cohomology $\bar{H}^3_{nr}(F(W),\mathbb Q/\mathbb Z(2))$.